Surely I'm not the only person that finds something *incredibly* wrong with this!?

Sorry but I see nothing but the misconception of const-correctness with the given example.
I hope you are aware of what you are asking.

For your particular example:
First you started by class, you should use struct. (sorry being lame here but happens...) Second why do mark something const if it is not, this is a wrong approach even in C++ which provides explicit "mutable".
How are you going to do caching anyways, if static array is not the case?

struct matrix {
        f32[4][4] m;
        f32 det_cache; // ?
        f32 mag_cache; // ?
        f32 whatever_cache; // ?
};

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